[4 marks sum]

Question 14 Marks

Factorise:$p ^2+\frac{1}{ p ^2}-3$

Answer

$p^2+\frac{1}{ p ^2}-3 $
$= P ^2+\frac{1}{ p ^2}-2-1 $
$ =\left( p ^2+\frac{1}{ p ^2}-2 \times p \times \frac{1}{ p }\right)-1$
$=\left( p -\frac{1}{ p }\right)^2-1^2$
$=\left( p -\frac{1}{ p }+1\right)\left( p -\frac{1}{ p }-1\right) .$

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Question 24 Marks

Factorise:$4x^4 + 25y^4 + 19x^2y^2$

Answer

$4x^4 + 25y^4 + 19x^2y^2$
$= 4x^4 + 25y^4 + 20x^2y^2 - x^2y^2$
$= (2x^2)^2 + (5y^2)^2 + 2 x (2x^2) x (5y^2) - x^2y^2$
$= [(2x^2)^2 + (5y^2) + 2 x (2x^2) x (5y^2)] -x^2y^2$
$= [2x^2 + 5y^2] - (xy)^2$
$= (2x^2 + 5y^2 - xy)(2x^2 - 5y + xy).$

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Question 34 Marks

Factorise the following:$(a^2 - b^2)(c^2 - d^2) - 4abcd$

Answer

$(a^2 - b^2)(c^2 - d^2) - 4abcd$
$= a^2c^2 - a^2d^2 - b^2c^2 + b^2d^2 - 4abcd$
$= a^2c^2 + b^2d^2 - 2abcd - a^2d^2 - b^2c^2 - 2abcd$
$= (a^2c^2 + b^2d^2 - 2abcd) - (a^2d^2 + b^2c^2 + 2abcd)$
$= (ac - bd)^2 - (ad + bc)^2$
$= [(ac - bd) + (ad + bc)][(ac - bd) - (ad + bc)]$
$= (ac - bd + ad + bc)(ac - bd - ad - bc).$

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Question 44 Marks

Factorise the following:$4xy - x^2 - 4y^2 + z^2$

Answer

$4xy - x^2 - 4y^2 + z^2$
$= z^2 - x^2 - 4y^2 + 4xy$
$= z^2 - (x^2 + 4y^2 - 4xy)$
$= z^2 - (x - 2y)^2$
$= [z - (x - 2y)][z + (x - 2y)]$
$= (z - x + 2y)(z + x - 2y).$

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Question 54 Marks

Factorise the following:$(x^2 + y^2 - z^2)^2 - 4x^2y^2$

Answer

$(x^2 + y^2 - z^2)^2 - 4x^2y^2$
$= (x^2 + y^2 - z^2)^2 - (2xy)^2$
$= (x^2 + y^2 - z^2 - 2xy)(x^2 + y^2 - z^2 + 2xy)$
$= [(x^2 + y^2 - 2xy) - z^2][(x^2 + y^2 + 2xy) - z^2]$
$= [(x - y)^2 - z^2][(x + y)^2 - z^2]$
$= [(x - y - z)(x - y + z)][(x + y - z)(x + y + z)]$
$= (x - y - z)(x - y + z)(x + y - z)(x + y + z).$

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Question 64 Marks

Factorise the following:$9(a - b)^2 - (a + b)^2$

Answer

$9(a - b)^2 - (a + b)^2$
$= [3(a - b)]^2 - (a + b)^2$
$= [3(a - b) - (a + b)][3(a - b) + (a + b)]$
$= (3a - 3b - a - b)(3a - 3b + a + b)$
$= (2a - 4b)(4a - 2b)$
$= 2(a - 2b)2(2a - b)$
$= 4(a - 2b)(2a - b).$

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Question 74 Marks

Factorise the following by the difference of two squares:$a(a - 1) - b(b - 1)$

Answer

$a(a - 1) - b(b - 1)$
$a^2 - a - b^2 + b$
$= a^2 - b^2 - a + b$
$= (a^2 - b^2) - (a - b)$
$= (a - b)(a + b) - (a - b)$
$= (a - b)(a + b - 1).$

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Question 84 Marks

Factorise the following by the difference of two squares:$16a^4 - 81b^4$

Answer

$16a^4 - 81b^4$
$= (4a^2)^2 - (9b^2)^2$
$= (4a^2 - 9b^2)(4a^2 + 9b^2)$
$= [(2a)^2 - (3b)^2](4a^2 + 9b^2)$
$= [(2a - 3b)(2a + 3b)](4a^2+ 9b^2)$
$= (2a - 3b)(2a + 3b)(4a^2 + 9b^2).$

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Question 94 Marks

Factorise the following:$2a^3 + 5a^2b - 12ab^2$

Answer

$2a^3 + 5a^2b - 12ab^2$
$= 2a^3 + 8a^2b - 3a^2b - 12ab^2$
$= 2a^2(a + 4b) - 3ab(a + 4b)$
$= (a + 4b)(2a^2 - 3ab)$
$= (a + 4b)a(2a - 3b)$
$= a(a + 4b)(2a - 3b).$

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Question 104 Marks

Factorise the following:$2 x^2+\frac{x}{6}-1$

Answer

$2 x^2+\frac{x}{6}-1$
$=\frac{1}{6}\left(12 x^2+x-6\right) $
$=\frac{1}{6}\left(12 x^2+9 x-8 x-6\right) $
$=\frac{1}{6}[3 x(4 x+3)-2(4 x+3)] $
$=\frac{1}{6}[(4 x+3)(3 x-2)]$
$ =\frac{1}{6}(4 x+3)(3 x-2) .$

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Question 114 Marks

Factorise the following:$6 - 5x + 5y + (x - y)^2$

Answer

$6 - 5x + 5y + (x - y)^2$
$= 6 - 5(x - y) + (x - )^2$
$= 6 - 3(x - y) - 2(x - y) + (x - y)^2$
$= 3[2 - (x - y)] - (x - y)[2 - (x - y)]$
$= 3(2 - x + y) - (x - y)(2 - x + y)$
$= (2 - x + y)(3 - x + y).$

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Question 124 Marks

Factorise the following:$12(2x - 3y)^2 - 2x + 3y - 1$

Answer

$12(2x - 3y)^2 - 1(2x - 3y) - 1$
$= 12a^2 - a - 1 [$Taking $(2x - 3y) = a]$
$= 12a^2 - 4a + 3a - 1$
$= 4a(3a - 1) + 1(3a - 1)$
$= (3a - 1)(4a + 1)$
$= [3(2x - 3y) - 1][4(2x - 3y) + 1]$
$= (6x - 9y - 1)(8x - 12y + 1).$

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Question 134 Marks

Factorise the following:$7(x - 2)^2 - 13(x - 2) - 2$

Answer

$7(x - 2)^2 - 13(x - 2) - 2$
$= 7(x - 2)^2 - 14(x - 2) + (x - 2) - 2$
$= 7(x - 2)(x - 2 - 2) + 1(x - 2 - 2)$
$= 7(x - 2)(x - 4) + 1(x - 4)$
$= (x - 4)[7(x - 2) + 1]$
$= (x - 4)(7x - 14 + 1)$
$= (x - 4)(7x - 13).$

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Question 144 Marks

Factorise the following:$(x + 4)^2 - 5xy - 20y - 6y^2$

Answer

$(x + 4)^2 - 5xy - 20y - 6y^2$
$= (x + 4)^2 - 5y(x + 4) - y^2$
$= (x + 4)^2 - 6y(x + 4) + y(x + 4) - 6y^2$
$= (x + 4)(x + 4 - 6y) + y(x + 4 - 6y)$
$= (x + 4 - 6y)(x + 4 + y)$
$= (x - 6y + 4)(x + y + 4).$

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Question 154 Marks

Factorise the following:$5 - 4(a - b) - 12(a - b)^2$

Answer

$5 - 4(a - b) - 12(a - b)^2$
$= 5 - 10(a - b) + 6(a - b) - 12(a - b)^2$
$= 5[1 - 2(a - b)] + 6(a - b)[1 - 2(a - b)]$
$= [5 + 6(a - b)][1 - 2(a - b)]$
$= (5 + 6a - 6b)(1 - 2a + 2b).$

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Question 164 Marks

Factorise the following:$x^3y^3 - 8x^2y^2 + 15xy$

Answer

$x^3y^3 - 8x^2y^2 + 15xy$
$= x^3y^3 - 3x^2y^2 - 5x^2y^2 + 15xy$
$= x^2y^2(xy - 3) - 5xy(xy - 3)$
$= (xy - 3)(x^2y^2 - 5xy)$
$= (xy - 3)xy(xy - 5)$
$= xy(xy - 3)(xy - 5).$

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Question 174 Marks

Factorise the following:$(2p + q)^2 - 10p - 5q - 6$

Answer

$(2p + q)^2 - 10p - 5q - 6$
$= (2p + q)^2 - (10p - 5q) - 6$
$= (2p + q)^2 - 5(2p + q) - 6$
$= (2p + q)^2 - 6(2p + q) + (2p + q) - 6$
$= (2p + q)(2p + q - 6) + 1(2p + q - 6)$
$= (2p + q - 6)(2p + q + 1).$

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Question 184 Marks

Factorise the following:$2x^3 + 5x^2y - 12xy^2$

Answer

$2x^3 + 5x^2y - 12xy^2$
$= 2x^3 + 8x^2y - 3x^2y - 12xy^2$
$= 2x^2(x + 4y) - 3xy(x + 4y)$
$= (x + 4y)(2x^2 - 3xy)$
$= (x + 4y)x(2x - 3y)$
$= x(x + 4y)(2x - 3y).$

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Question 194 Marks

Factorise the following by taking out the common factors:$p(p^2 + q^2 - r^2) + q(r^2 - q^2 -p^2) - r(p^2 + q^2 - r^2)$

Answer

$p\left(p^2+q^2-r^2\right)+q\left(r^2-q^2-p^2\right)-r\left(p^2+q^2-r^2\right) $
$=p\left(p^2+q^2-r^2\right)-q\left(-r^2+q^2+p^2\right)-r\left(p^2+q^2-r^2\right) $
$=p\left(p^2+q^2-r^2\right)-q\left(p^2+q^2-r^2\right)-r\left(p^2+q^2-r^2\right)$
Here, the common factor is $\left(p^2+q^2-r^2\right)$.
Dividing throughout by $\left(p^2+q^2-r^2\right)$
$\frac{p\left(p^2+q^2-r^2\right)}{p^2+q^2-r^2}-\frac{q\left(p^2+q^2-r^2\right)}{\left(p^2+q^2-r^2\right)}-\frac{r\left(p^2+q^2-r^2\right)}{\left(p^2+q^2-r^2\right)} $
$=p-q-r $
$\therefore p\left(p^2+q^2-r^2\right)+q\left(r^2-q^2-p^2\right)-r\left(p^2+q^2-r^2\right) $
$=\left(p^2+q^2-r^2\right)(p-q-r) .$

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Question 204 Marks

Factorise the following by taking out the common factors:$(mx + ny)^2 + (nx - my)^2$

Answer

$(m x+n y)^2+(n x-m y)^2 $
$=m^2 x^2+n^2 y^2+2 m n x y+n^2 x^2+m^2 y^2-2 m n x y $
$=m^2 x^2+n^2 y^2+n^2 x^2+m^2 y^2 $
$=m^2 x^2+n^2 x^2+m^2 y^2+n^2 y^2 $
$=x^2\left(m^2+n^2\right)+y^2\left(m^2+n^2\right)$
Here, the common factor is $\left(m^2+n^2\right)$.
Dividing throughout by $\left(m^2+n^2\right)$, we get
$\frac{x^2\left(m^2+n^2\right)}{\left(m^2+n^2\right)}+\frac{y^2\left(m^2+n^2\right)}{\left(m^2+n^2\right)} $
$=x^2+y^2 $
$\therefore(m x+n y)^2+(n x-m y)^2 $
$=\left(m^2+n^2\right)\left(x^2+y^2\right) .$

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MATHEMATICS [4 marks sum]

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